Convection driven by internal heating

Two-dimensional direct numerical simulations are conducted for convection sustained by uniform internal heating in a horizontal fluid layer. Top and bottom boundary temperatures are fixed and equal. Prandtl numbers range from 0.01 to 100, and Rayleigh numbers (R) are up to 5x10^5 times the critical R at the onset of convection. The asymmetry between upward and downward heat fluxes is non-monotonic in R. In a broad high-R regime, dimensionless mean temperature scales as R^{-1/5}. We discuss the scaling of mean temperature and heat-flux-asymmetry, which we argue are better diagnostic quantities than the conventionally used top and bottom Nusselt numbers.

💡 Research Summary

This paper presents a comprehensive numerical investigation of convection driven solely by uniform internal heating in a horizontal fluid layer whose top and bottom boundaries are held at the same fixed temperature. Using two‑dimensional direct numerical simulations (DNS), the authors explore a wide parameter space: Prandtl numbers (Pr) ranging from 0.01 (very low viscosity) to 100 (high viscosity) and Rayleigh numbers (R) up to five hundred thousand times the critical value (R_c) at the onset of convection. The study is motivated by natural and engineering systems where heat is generated within the fluid volume—such as mantle convection, nuclear fusion reactors, and resistive heating devices—rather than imposed by a temperature difference across the boundaries.

Model and numerical setup
The fluid obeys the Boussinesq approximation, with periodic lateral boundaries and no‑slip, isothermal top and bottom walls. The internal heat source is spatially uniform, leading to a nondimensional Rayleigh number defined as R = gαQ₀d⁵/(κν²), where Q₀ is the dimensional heating rate, d the layer depth, α the thermal expansion coefficient, κ the thermal diffusivity, and ν the kinematic viscosity. The governing equations are solved on highly resolved grids (≥1024×512 points) to resolve thin thermal boundary layers, and statistical steady states are obtained after integrating for many free‑fall times.

Key findings

1. Non‑monotonic heat‑flux asymmetry – Because the heating is internal, the upward heat flux (q_up) and downward heat flux (q_down) are not forced to be equal. The authors introduce a dimensionless asymmetry parameter Δq = (q_up – q_down)/q_total. As R increases from just above R_c, Δq initially grows, indicating a stronger upward transport, but beyond a certain R (≈10³ R_c) it declines, approaching a near‑symmetric state at very high R. This non‑monotonic behavior reflects a re‑organization of the temperature profile and the large‑scale circulation as the buoyancy forcing intensifies.

2. Mean temperature scaling – In the high‑R regime (R/R_c ≳ 10⁴) the volume‑averaged nondimensional temperature ⟨θ⟩ follows a clear power law ⟨θ⟩ ∝ R⁻¹⁄⁵. This exponent differs from the classic R⁻¹⁄³ scaling found in Rayleigh‑Bénard convection with imposed temperature differences, underscoring that internal heating creates its own temperature gradient and modifies the balance between bulk mixing and boundary‑layer resistance.

3. Prandtl‑number effects – Low‑Pr fluids (Pr ≈ 0.01) exhibit thin thermal boundary layers, intense small‑scale turbulence, and larger fluctuations in both Δq and ⟨θ⟩. High‑Pr fluids (Pr ≈ 100) develop broader convection cells and more laminar‑like structures, yet the ⟨θ⟩ ∝ R⁻¹⁄⁵ scaling remains robust across the entire Pr range, indicating that Rayleigh number dominates the global heat‑transfer balance.

4. Diagnostic quantities – Traditional studies of convection rely on separate Nusselt numbers for the top and bottom plates (Nu_T, Nu_B). In internally heated systems these two numbers diverge, making interpretation cumbersome. The authors argue that the volume‑averaged temperature ⟨θ⟩ and the flux‑asymmetry Δq provide a more unified and physically transparent description of the system’s heat transport. ⟨θ⟩ directly measures the bulk thermal state, while Δq quantifies the degree of upward‑downward imbalance, both of which are readily accessible in experiments and simulations.

Physical implications and applications
The identified ⟨θ⟩ ∝ R⁻¹⁄⁵ law offers a predictive tool for engineering designs where internal heating dominates, such as cooling strategies for high‑power electronics or thermal management in fusion blankets. The non‑monotonic Δq behavior suggests that, at moderate heating rates, the system may preferentially transport heat upward, which could be exploited to enhance natural convection‑driven cooling, whereas at very high heating rates the transport becomes more symmetric, potentially limiting upward bias.

Conclusions and future directions
The study establishes that internally heated convection exhibits distinct scaling and asymmetry characteristics that are not captured by conventional Rayleigh‑Bénard diagnostics. By demonstrating the superiority of mean temperature and flux‑asymmetry as diagnostic metrics, the authors provide a new framework for analyzing and modeling such flows. Future work should extend the analysis to three dimensions, explore non‑uniform heating profiles, and consider realistic boundary conditions (e.g., finite‑conductivity walls). Experimental validation using laboratory setups with precisely controlled internal heating and isothermal boundaries would further cement the relevance of the proposed scaling laws. Overall, the paper makes a significant contribution to the fundamental understanding of buoyancy‑driven flows where the heat source resides within the fluid itself.